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 Coth

 http://functions.wolfram.com/01.22.21.0279.01

 Input Form

 Integrate[Coth[c z] Sqrt[(a + b Coth[c z]^2)^3], z] == -(2 Sqrt[(a + b Coth[c z]^2)^3] ((-(2 a + b)) Sqrt[-a + b + (a + b) Cosh[2 c z]] + 2 (a + b) Cosh[2 c z] Sqrt[-a + b + (a + b) Cosh[2 c z]] - 3 Sqrt[2] Log[Sqrt[-a + b + (a + b) Cosh[2 c z]] + Sqrt[2] Sqrt[(a + b) Sinh[c z]^2]] ((a + b) Sinh[c z]^2)^(3/2)))/ (3 c (-a + b + (a + b) Cosh[2 c z])^(3/2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b"]], ")"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]]]]]]

 MathML Form

 coth ( c z ) ( a + b coth 2 ( c z ) ) 3 z - ( 2 ( b coth 2 ( c z ) + a ) 3 ( - 3 2 log ( - a + b + ( a + b ) cosh ( 2 c z ) + 2 ( a + b ) sinh 2 ( c z ) ) ( ( a + b ) sinh 2 ( c z ) ) 3 / 2 - ( 2 a + b ) - a + b + ( a + b ) cosh ( 2 c z ) + 2 ( a + b ) cosh ( 2 c z ) - a + b + ( a + b ) cosh ( 2 c z ) ) ) / ( 3 c ( - a + b + ( a + b ) cosh ( 2 c z ) ) 3 / 2 ) z c z a b c z 2 3 1 2 -1 2 b c z 2 a 3 1 2 -3 2 1 2 -1 a b a b 2 c z 1 2 2 1 2 a b c z 2 1 2 a b c z 2 3 2 -1 2 a b -1 a b a b 2 c z 1 2 2 a b 2 c z -1 a b a b 2 c z 1 2 3 c -1 a b a b 2 c z 3 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "3"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["2", " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b"]], ")"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "-", RowBox[List["3", " ", SqrtBox["2"], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]], RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2002-12-18